Exercise 15, § 1

Soit X a locally compact topological space, $\mu$ a bounded positive measure on X with support X, and $N\in \mathscr{C}_b(X\times X)$. a) There exists an endomorphism $u$ of $\mathscr{C}_b(X)$ such that

for every function $f\in \mathscr{C}_b(X)$ and every $y\in X$. We suppose in what follows that $u$ is compact; this is the case for example if the space X is compact.

b) The image of $u_N$ is contained in $\mathscr{C}_b(X)$ and $u_N$ defines a continuous linear mapping from $L^2_{\mathbf{C}}(X)$ into $\mathscr{C}_b(X)$.

c) Let $f\in L^2_{\mathbf{C}}(X)$ and $\lambda \in \mathbf{C}^*$ be such that $u_N(f) =\lambda f$. Then $f\in \mathscr{C}_b(X)$.

d) There exists a countable family $(f_i)_{i\in I}$ in $\mathscr{C}_b(X)$ and a family $(\lambda_i)_{i\in I}$ of nonzero real numbers such that for every $f\in L^2_{\mathbf{C}}$(X), one has

in $\mathscr{C}_b(X)$.

e) We suppose that $u_N$ is a positive endomorphism of $L^2_{\mathbf{C}}(X)$. For every $i\in I$, we denote by $h_i\in \mathscr{C}_b(X\times X)$ the function defined by $h_i(x, y) =\overline{f_i(x)}f_i(y)$. Then

in $\mathscr{C}(X\times X)$ endowed with the topology of compact convergence. (Prove that the series converges using Dini’s theorem (TG, X, p. 34, th. 1), then prove the equality.)